Julia sets converging to the unit disk

Abstract:

We consider the family of rational maps $F_\lambda (z) = z^n + \lambda /z^d$, where $n,d \geq 2$ and $\lambda$ is small. If $\lambda$ is equal to 0, the limiting map is $F_0(z)=z^n$ and the Julia set is the unit circle. We investigate the behavior of the Julia sets of $F_\lambda$ when $\lambda$ tends to 0, obtaining two very different cases depending on $n$ and $d$. The first case occurs when $n=d=2$; here the Julia sets of $F_\lambda$ converge as sets to the closed unit disk. In the second case, when one of $n$ or $d$ is larger than $2$, there is always an annulus of some fixed size in the complement of the Julia set, no matter how small $|\lambda |$ is.